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Theorem monpropd 15009
Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same monomorphisms. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
monpropd.3  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
monpropd.4  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
monpropd.c  |-  ( ph  ->  C  e.  Cat )
monpropd.d  |-  ( ph  ->  D  e.  Cat )
Assertion
Ref Expression
monpropd  |-  ( ph  ->  (Mono `  C )  =  (Mono `  D )
)

Proof of Theorem monpropd
Dummy variables  a 
b  c  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . . . . . . . . . . . 12  |-  ( Base `  C )  =  (
Base `  C )
2 eqid 2443 . . . . . . . . . . . 12  |-  ( Hom  `  C )  =  ( Hom  `  C )
3 eqid 2443 . . . . . . . . . . . 12  |-  ( Hom  `  D )  =  ( Hom  `  D )
4 monpropd.3 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
54ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
65ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  -> 
( Hom f  `  C )  =  ( Hom f  `  D ) )
7 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  -> 
c  e.  ( Base `  C ) )
8 simp-4r 768 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  -> 
a  e.  ( Base `  C ) )
91, 2, 3, 6, 7, 8homfeqval 14969 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  -> 
( c ( Hom  `  C ) a )  =  ( c ( Hom  `  D )
a ) )
10 eqid 2443 . . . . . . . . . . . 12  |-  (comp `  C )  =  (comp `  C )
11 eqid 2443 . . . . . . . . . . . 12  |-  (comp `  D )  =  (comp `  D )
124ad5antr 733 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a ( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c ( Hom  `  C ) a ) )  ->  ( Hom f  `  C
)  =  ( Hom f  `  D ) )
13 monpropd.4 . . . . . . . . . . . . 13  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
1413ad5antr 733 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a ( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c ( Hom  `  C ) a ) )  ->  (compf `  C )  =  (compf `  D ) )
15 simplr 755 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a ( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c ( Hom  `  C ) a ) )  ->  c  e.  ( Base `  C )
)
16 simp-5r 770 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a ( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c ( Hom  `  C ) a ) )  ->  a  e.  ( Base `  C )
)
17 simp-4r 768 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a ( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c ( Hom  `  C ) a ) )  ->  b  e.  ( Base `  C )
)
18 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a ( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c ( Hom  `  C ) a ) )  ->  g  e.  ( c ( Hom  `  C ) a ) )
19 simpllr 760 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a ( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c ( Hom  `  C ) a ) )  ->  f  e.  ( a ( Hom  `  C ) b ) )
201, 2, 10, 11, 12, 14, 15, 16, 17, 18, 19comfeqval 14980 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a ( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c ( Hom  `  C ) a ) )  ->  ( f
( <. c ,  a
>. (comp `  C )
b ) g )  =  ( f (
<. c ,  a >.
(comp `  D )
b ) g ) )
219, 20mpteq12dva 4514 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  -> 
( g  e.  ( c ( Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) )  =  ( g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) )
2221cnveqd 5168 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  ->  `' ( g  e.  ( c ( Hom  `  C ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  C )
b ) g ) )  =  `' ( g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) )
2322funeqd 5599 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  -> 
( Fun  `' (
g  e.  ( c ( Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) )  <->  Fun  `' ( g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) ) )
2423ralbidva 2879 . . . . . . 7  |-  ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
( Hom  `  C ) b ) )  -> 
( A. c  e.  ( Base `  C
) Fun  `' (
g  e.  ( c ( Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) )  <->  A. c  e.  ( Base `  C
) Fun  `' (
g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) ) )
2524rabbidva 3086 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  { f  e.  ( a ( Hom  `  C )
b )  |  A. c  e.  ( Base `  C ) Fun  `' ( g  e.  ( c ( Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) ) }  =  { f  e.  ( a ( Hom  `  C ) b )  |  A. c  e.  ( Base `  C
) Fun  `' (
g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } )
26 simplr 755 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  a  e.  ( Base `  C
) )
27 simpr 461 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  b  e.  ( Base `  C
) )
281, 2, 3, 5, 26, 27homfeqval 14969 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  (
a ( Hom  `  C
) b )  =  ( a ( Hom  `  D ) b ) )
294homfeqbas 14968 . . . . . . . . 9  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
3029ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  ( Base `  C )  =  ( Base `  D
) )
3130raleqdv 3046 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  ( A. c  e.  ( Base `  C ) Fun  `' ( g  e.  ( c ( Hom  `  D ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  D )
b ) g ) )  <->  A. c  e.  (
Base `  D ) Fun  `' ( g  e.  ( c ( Hom  `  D ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  D )
b ) g ) ) ) )
3228, 31rabeqbidv 3090 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  { f  e.  ( a ( Hom  `  C )
b )  |  A. c  e.  ( Base `  C ) Fun  `' ( g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) }  =  { f  e.  ( a ( Hom  `  D ) b )  |  A. c  e.  ( Base `  D
) Fun  `' (
g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } )
3325, 32eqtrd 2484 . . . . 5  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  { f  e.  ( a ( Hom  `  C )
b )  |  A. c  e.  ( Base `  C ) Fun  `' ( g  e.  ( c ( Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) ) }  =  { f  e.  ( a ( Hom  `  D ) b )  |  A. c  e.  ( Base `  D
) Fun  `' (
g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } )
34333impa 1192 . . . 4  |-  ( (
ph  /\  a  e.  ( Base `  C )  /\  b  e.  ( Base `  C ) )  ->  { f  e.  ( a ( Hom  `  C ) b )  |  A. c  e.  ( Base `  C
) Fun  `' (
g  e.  ( c ( Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) ) }  =  { f  e.  ( a ( Hom  `  D ) b )  |  A. c  e.  ( Base `  D
) Fun  `' (
g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } )
3534mpt2eq3dva 6346 . . 3  |-  ( ph  ->  ( a  e.  (
Base `  C ) ,  b  e.  ( Base `  C )  |->  { f  e.  ( a ( Hom  `  C
) b )  | 
A. c  e.  (
Base `  C ) Fun  `' ( g  e.  ( c ( Hom  `  C ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  C )
b ) g ) ) } )  =  ( a  e.  (
Base `  C ) ,  b  e.  ( Base `  C )  |->  { f  e.  ( a ( Hom  `  D
) b )  | 
A. c  e.  (
Base `  D ) Fun  `' ( g  e.  ( c ( Hom  `  D ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  D )
b ) g ) ) } ) )
36 mpt2eq12 6342 . . . 4  |-  ( ( ( Base `  C
)  =  ( Base `  D )  /\  ( Base `  C )  =  ( Base `  D
) )  ->  (
a  e.  ( Base `  C ) ,  b  e.  ( Base `  C
)  |->  { f  e.  ( a ( Hom  `  D ) b )  |  A. c  e.  ( Base `  D
) Fun  `' (
g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } )  =  ( a  e.  ( Base `  D
) ,  b  e.  ( Base `  D
)  |->  { f  e.  ( a ( Hom  `  D ) b )  |  A. c  e.  ( Base `  D
) Fun  `' (
g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } ) )
3729, 29, 36syl2anc 661 . . 3  |-  ( ph  ->  ( a  e.  (
Base `  C ) ,  b  e.  ( Base `  C )  |->  { f  e.  ( a ( Hom  `  D
) b )  | 
A. c  e.  (
Base `  D ) Fun  `' ( g  e.  ( c ( Hom  `  D ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  D )
b ) g ) ) } )  =  ( a  e.  (
Base `  D ) ,  b  e.  ( Base `  D )  |->  { f  e.  ( a ( Hom  `  D
) b )  | 
A. c  e.  (
Base `  D ) Fun  `' ( g  e.  ( c ( Hom  `  D ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  D )
b ) g ) ) } ) )
3835, 37eqtrd 2484 . 2  |-  ( ph  ->  ( a  e.  (
Base `  C ) ,  b  e.  ( Base `  C )  |->  { f  e.  ( a ( Hom  `  C
) b )  | 
A. c  e.  (
Base `  C ) Fun  `' ( g  e.  ( c ( Hom  `  C ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  C )
b ) g ) ) } )  =  ( a  e.  (
Base `  D ) ,  b  e.  ( Base `  D )  |->  { f  e.  ( a ( Hom  `  D
) b )  | 
A. c  e.  (
Base `  D ) Fun  `' ( g  e.  ( c ( Hom  `  D ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  D )
b ) g ) ) } ) )
39 eqid 2443 . . 3  |-  (Mono `  C )  =  (Mono `  C )
40 monpropd.c . . 3  |-  ( ph  ->  C  e.  Cat )
411, 2, 10, 39, 40monfval 15004 . 2  |-  ( ph  ->  (Mono `  C )  =  ( a  e.  ( Base `  C
) ,  b  e.  ( Base `  C
)  |->  { f  e.  ( a ( Hom  `  C ) b )  |  A. c  e.  ( Base `  C
) Fun  `' (
g  e.  ( c ( Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) ) } ) )
42 eqid 2443 . . 3  |-  ( Base `  D )  =  (
Base `  D )
43 eqid 2443 . . 3  |-  (Mono `  D )  =  (Mono `  D )
44 monpropd.d . . 3  |-  ( ph  ->  D  e.  Cat )
4542, 3, 11, 43, 44monfval 15004 . 2  |-  ( ph  ->  (Mono `  D )  =  ( a  e.  ( Base `  D
) ,  b  e.  ( Base `  D
)  |->  { f  e.  ( a ( Hom  `  D ) b )  |  A. c  e.  ( Base `  D
) Fun  `' (
g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } ) )
4638, 41, 453eqtr4d 2494 1  |-  ( ph  ->  (Mono `  C )  =  (Mono `  D )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   {crab 2797   <.cop 4020    |-> cmpt 4495   `'ccnv 4988   Fun wfun 5572   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   Basecbs 14509   Hom chom 14585  compcco 14586   Catccat 14938   Hom f chomf 14940  compfccomf 14941  Monocmon 15000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-homf 14944  df-comf 14945  df-mon 15002
This theorem is referenced by:  oppcepi  15011
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